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Diffstat (limited to 'gcs')
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| -rw-r--r-- | gcs/n3.lyx | 4480 |
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@@ -175,5 +175,19 @@ filename "n2.lyx" \end_layout +\begin_layout Chapter +Curvatura de superficies +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n3.lyx" + +\end_inset + + +\end_layout + \end_body \end_document diff --git a/gcs/n3.lyx b/gcs/n3.lyx new file mode 100644 index 0000000..02ffa54 --- /dev/null +++ b/gcs/n3.lyx @@ -0,0 +1,4480 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Section +Orientación +\end_layout + +\begin_layout Standard +Dada una superficie regular +\begin_inset Formula $S$ +\end_inset + +, un +\series bold +campo de vectores +\series default + sobre +\begin_inset Formula $S$ +\end_inset + + es una función +\begin_inset Formula $\xi:S\to\mathbb{R}^{3}$ +\end_inset + +, y es +\series bold +tangente +\series default + si +\begin_inset Formula $\xi(p)\in T_{p}S$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + +, +\series bold +normal +\series default + si +\begin_inset Formula $\xi(p)\in(T_{p}S)^{\bot}$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + + y +\series bold +unitario +\series default + si +\begin_inset Formula $|\xi(p)|=1$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + +. + Llamamos +\begin_inset Formula $\mathfrak{X}(S)$ +\end_inset + + al conjunto de campos de vectores tangentes sobre +\begin_inset Formula $S$ +\end_inset + + y +\begin_inset Formula $\mathfrak{X}(S)^{\bot}$ +\end_inset + + al conjunto de campos de vectores normales sobre +\begin_inset Formula $S$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Una +\series bold +orientación +\series default + de una superficie regular +\begin_inset Formula $S$ +\end_inset + + es un campo de vectores diferenciable, normal y unitario sobre +\begin_inset Formula $S$ +\end_inset + +. + +\begin_inset Formula $S$ +\end_inset + + es +\series bold +orientable +\series default + si admite una orientación, si y sólo si existe un campo +\begin_inset Formula $\xi$ +\end_inset + + normal y diferenciable sobre +\begin_inset Formula $S$ +\end_inset + + que no se anula en ningún punto, pues las orientaciones son de esta forma + y, dado +\begin_inset Formula $\xi$ +\end_inset + +, basta tomar la orientación +\begin_inset Formula $N(p):=\xi(p)/|\xi(p)|$ +\end_inset + +. + Una orientación +\begin_inset Formula $N$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + da a cada +\begin_inset Formula $p\in S$ +\end_inset + + un sentido de giro para +\begin_inset Formula $T_{p}S$ +\end_inset + + dado por el producto vectorial en +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + +. + +\begin_inset Formula $S$ +\end_inset + + está orientada cuando se ha escogido una orientación concreta, en cuyo + caso dicha orientación es su +\series bold +aplicación de Gauss +\series default +. +\end_layout + +\begin_layout Standard +Ejemplos: +\begin_inset Note Comment +status open + +\begin_layout Enumerate +La banda de Möbius se puede expresar como la imagen de +\begin_inset Formula $X:\mathbb{R}\times(-1,1)\to\mathbb{R}^{3}$ +\end_inset + + dada por +\begin_inset Formula +\[ +X(u,v):=\left((2-v\sin\tfrac{u}{2})\sin u,(2-v\sin\tfrac{u}{2})\cos u,v\cos\tfrac{u}{2}\right). +\] + +\end_inset + +Esta es una superficie regular no orientable. +\end_layout + +\begin_deeper +\begin_layout Plain Layout +Claramente +\begin_inset Formula $X$ +\end_inset + + es diferenciable, y es inyectiva en +\begin_inset Formula $U_{1}:=(0,2\pi)\times(-1,1)$ +\end_inset + + y en +\begin_inset Formula $U_{2}:=(-\pi,\pi)\times(-1,1)$ +\end_inset + +. + Su diferencial es +\begin_inset Formula +\[ +dX(u,v)\equiv\begin{pmatrix}-\frac{v}{2}\cos\frac{u}{2}\sin u+(2-v\sin\frac{u}{2})\cos u & -\sin\frac{u}{2}\sin u\\ +-\frac{v}{2}\cos\frac{u}{2}\cos u-(2-v\sin\frac{u}{2})\sin u & -\sin\frac{u}{2}\cos u\\ +-\frac{v}{2}\sin\frac{u}{2} & \cos\frac{u}{2} +\end{pmatrix}, +\] + +\end_inset + +y el determinante de las dos primeras filas es +\begin_inset Formula +\[ +-\sin\frac{u}{2}\left(-\frac{v}{2}\cos\frac{u}{2}\begin{vmatrix}\sin u & \sin u\\ +\cos u & \cos u +\end{vmatrix}+\left(2-v\sin\frac{u}{2}\right)\begin{vmatrix}\cos u & \sin u\\ +-\sin u & \cos u +\end{vmatrix}\right)=-\sin\frac{u}{2}\left(2-v\sin\frac{u}{2}\right), +\] + +\end_inset + +lo que solo se anula cuando +\begin_inset Formula $u\in\{2k\pi\}_{k\in\mathbb{Z}}$ +\end_inset + +, pero en tal caso +\begin_inset Formula +\[ +dX(u,v)\equiv\begin{pmatrix}2 & 0\\ +-\frac{v}{2} & 0\\ +0 & 1 +\end{pmatrix} +\] + +\end_inset + +y el determinante de la submatriz resultante de quitar la segunda fila es + +\begin_inset Formula $2\neq0$ +\end_inset + +. + Esto prueba que la banda de Möbius es una superficie. +\end_layout + +\end_deeper +\end_inset + + +\end_layout + +\begin_layout Enumerate +El plano +\begin_inset Formula $p_{0}+\langle v\rangle^{\bot}\subseteq\mathbb{R}^{3}$ +\end_inset + + admite la orientación +\begin_inset Formula $N(p):=v/|v|$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Dados +\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ +\end_inset + + +\begin_inset Formula ${\cal C}^{2}$ +\end_inset + + y un valor regular +\begin_inset Formula $c$ +\end_inset + + de +\begin_inset Formula $f$ +\end_inset + +, la superficie de nivel +\begin_inset Formula $S:=f^{-1}(c)$ +\end_inset + + admite la orientación +\begin_inset Formula +\[ +N(p):=\frac{\nabla f(p)}{|\nabla f(p)|}, +\] + +\end_inset + + donde +\begin_inset Formula $\nabla f(p):=(\frac{\partial f}{\partial x}(p),\frac{\partial f}{\partial y}(p),\frac{\partial f}{\partial z}(p))$ +\end_inset + + es el +\series bold +gradiente +\series default + de +\begin_inset Formula $f$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $\alpha:=(x,y,z):I\to S$ +\end_inset + + una curva diferenciable con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $v:=\alpha'(0)\in T_{p}S$ +\end_inset + +, para +\begin_inset Formula $t\in I$ +\end_inset + + es +\begin_inset Formula $f(\alpha(t))=c$ +\end_inset + + por ser +\begin_inset Formula $\alpha(t)\in S$ +\end_inset + +, luego derivando, +\begin_inset Formula $\frac{\partial f}{\partial x}(\alpha(t))x'(t)+\frac{\partial f}{\partial y}(\alpha(t))y'(t)+\frac{\partial f}{\partial z}(\alpha(t))z'(t)=0$ +\end_inset + + y +\begin_inset Formula $\nabla f(p)\bot v$ +\end_inset + +. + Además, +\begin_inset Formula $\nabla f(p)\neq0$ +\end_inset + + porque +\begin_inset Formula $p\in S=f^{-1}(c)$ +\end_inset + + y +\begin_inset Formula $c$ +\end_inset + + es un valor regular de +\begin_inset Formula $f$ +\end_inset + +, y claramente +\begin_inset Formula $\nabla f$ +\end_inset + + es diferenciable. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\mathbb{S}^{2}(r)$ +\end_inset + + admite la orientación +\begin_inset Formula $N(p)=\frac{1}{r}p$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}+z^{2}$ +\end_inset + +, +\begin_inset Formula $r^{2}$ +\end_inset + + es un valor regular de +\begin_inset Formula $f$ +\end_inset + + y +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + + es la superficie de nivel +\begin_inset Formula $\{p:f(p)=r^{2}\}$ +\end_inset + +, luego admite la orientación +\begin_inset Formula +\[ +N(x,y,z)=\frac{\nabla f(x,y,z)}{|\nabla f(x,y,z)|}=\frac{(2x,2y,2z)}{|(2x,2y,2z)|}=\frac{(x,y,z)}{|(x,y,z)|}=\frac{1}{r}(x,y,z). +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +El cilindro +\begin_inset Formula $\{x^{2}+y^{2}=r^{2}\}$ +\end_inset + + admite la orientación +\begin_inset Formula $N(x,y,z)=(x,y,0)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Es una superficie de nivel y tiene pues orientación +\begin_inset Formula $N(p)=\frac{(2x,2y,0)}{|(2x,2y,0)|}=\frac{(x,y,0)}{|(x,y,0)|}=\frac{1}{r}(x,y,0)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Dada +\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + + diferenciable en el abierto +\begin_inset Formula $U$ +\end_inset + +, el grafo +\begin_inset Formula $S:=\{(x,y,f(x,y))\}_{x,y\in U}$ +\end_inset + + admite la orientación +\begin_inset Formula +\[ +N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}(u,v). +\] + +\end_inset + +Dada la parametrización +\begin_inset Formula $(U,X)$ +\end_inset + + con +\begin_inset Formula $X(u,v):=(u,v,f(u,v))$ +\end_inset + +, +\begin_inset Formula $X_{u}=(1,0,f_{u})$ +\end_inset + + y +\begin_inset Formula $X_{v}=(0,1,f_{v})$ +\end_inset + +, y +\begin_inset Formula $X_{u}\wedge X_{v}=(-f_{u},-f_{v},1)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Las superficies orientables tienen exactamente dos orientaciones, una opuesta + de la otra. +\end_layout + +\begin_layout Standard +Dos cartas +\begin_inset Formula $(U,X)$ +\end_inset + + y +\begin_inset Formula $(U',X')$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + son +\series bold +compatibles +\series default + si +\begin_inset Formula $V:=X(U)$ +\end_inset + + y +\begin_inset Formula $V':=X'(U')$ +\end_inset + + son disjuntos o +\begin_inset Formula $\det(Jh)>0$ +\end_inset + +, donde +\begin_inset Formula $h:X^{-1}(V')\to(X')^{-1}(V)$ +\end_inset + + es el cambio de coordenadas de +\begin_inset Formula $V$ +\end_inset + + a +\begin_inset Formula $V'$ +\end_inset + +. + Un +\series bold +atlas +\series default + para +\begin_inset Formula $S$ +\end_inset + + es una familia +\begin_inset Formula $\{(U_{i},X_{i})\}_{i\in I}$ +\end_inset + + de cartas tales que +\begin_inset Formula $\bigcup_{i\in I}X_{i}(U_{i})=S$ +\end_inset + +. + Entonces una superficie es orientable si y sólo si existe un atlas cuyas + cartas son compatibles. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Sean +\begin_inset Formula ${\cal A}:=\{(U_{i},X_{i})\}_{i\in I}$ +\end_inset + + un atlas de cartas compatibles en +\begin_inset Formula $S$ +\end_inset + +, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $(U,X)\in{\cal A}(I)$ +\end_inset + + con +\begin_inset Formula $p\in X(U)$ +\end_inset + + y +\begin_inset Formula $N:X(U)\to\mathbb{R}^{3}$ +\end_inset + + dado por +\begin_inset Formula +\[ +N(X(u,v)):=N(u,v):=\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}(u,v), +\] + +\end_inset + + +\begin_inset Formula $N$ +\end_inset + + está bien definido y es diferenciable, normal y unitario. + Sean ahora +\begin_inset Formula $(\overline{U},\overline{X})\in{\cal A}(I)$ +\end_inset + + con +\begin_inset Formula $p\in\overline{X}(\overline{U})$ +\end_inset + +, +\begin_inset Formula $\overline{N}(\overline{X}(u,v)):=\overline{N}(u,v):=\frac{\overline{X}_{u}\cap\overline{X}_{v}}{|\overline{X}_{u}\cap\overline{X}_{v}|}(u,v)$ +\end_inset + + y +\begin_inset Formula $h$ +\end_inset + + el cambio de coordenadas de +\begin_inset Formula $(U,X)$ +\end_inset + + a +\begin_inset Formula $(\overline{U},\overline{X})$ +\end_inset + +, para +\begin_inset Formula $(u,v)\in X^{-1}(V_{0})$ +\end_inset + +, +\begin_inset Formula +\[ +dX(u,v)=d(\overline{X}\circ h)(u,v)=d\overline{X}(h(u,v))\circ dh(u,v), +\] + +\end_inset + + luego +\begin_inset Formula +\[ +N(u,v)=\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}=\frac{\det(Jh(u,v))}{|\det(Jh(u,v))|}\frac{\overline{X}_{u}\wedge\overline{X}_{v}}{|\overline{X}_{u}\wedge\overline{X}_{v}|}(h(u,v))\overset{Jh(u,v)>0}{=}\overline{N}(u,v), +\] + +\end_inset + + de modo que +\begin_inset Formula $N(p)$ +\end_inset + + es diferenciable, normal, unitario y no depende de la carta del atlas escogida. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Sea +\begin_inset Formula $N$ +\end_inset + + una orientación de +\begin_inset Formula $S$ +\end_inset + +, para toda carta +\begin_inset Formula $(U,X)$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + es +\begin_inset Formula $N(X(q))=\pm\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}(q)$ +\end_inset + + para todo +\begin_inset Formula $q\in U$ +\end_inset + +. + Entonces, para +\begin_inset Formula $p\in S$ +\end_inset + +, podemos tomar una carta +\begin_inset Formula $(U_{p},X_{p})$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + con +\begin_inset Formula $N(X(q))=\frac{(X_{p})_{u}\wedge(X_{p})_{v}}{|(X_{p})_{u}\wedge(X_{p})_{v}|}(q)$ +\end_inset + + para +\begin_inset Formula $q\in U$ +\end_inset + +, pues si el normal fuese el opuesto basta cambiar +\begin_inset Formula $X_{p}(u,v)$ +\end_inset + + por +\begin_inset Formula $X_{p}(v,u)$ +\end_inset + + y +\begin_inset Formula $U_{p}$ +\end_inset + + por +\begin_inset Formula $\{(u,v)\}_{(v,u)\in U}$ +\end_inset + +, y el resultado se tiene por la antisimetría del producto vectorial. + Con esto, dados +\begin_inset Formula $a,b\in S$ +\end_inset + + con +\begin_inset Formula $V:=X_{a}(U_{a})\cap X_{b}(U_{b})\neq\emptyset$ +\end_inset + +, queremos ver que el determinante del cambio de coordenadas +\begin_inset Formula $h:X_{a}^{-1}(V)\to X_{b}^{-1}(V)$ +\end_inset + + de +\begin_inset Formula $(U_{a},X_{a})$ +\end_inset + + a +\begin_inset Formula $(U_{b},X_{b})$ +\end_inset + + tiene jacobiano con determinante positivo. + En efecto, +\begin_inset Formula $\det(Jh)$ +\end_inset + + debe ser no nulo, pero si fuera negativo, para un +\begin_inset Formula $p\in V$ +\end_inset + +, sean +\begin_inset Formula $q_{a}:=X_{a}^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $q_{b}:=X_{b}^{-1}(p)$ +\end_inset + +, entonces +\begin_inset Formula +\[ +N(p)=\frac{X_{au}\wedge X_{av}}{|X_{au}\wedge X_{av}|}(q_{a})=\frac{\det(Jh)}{|\det(Jh)|}\frac{X_{bu}\wedge X_{bv}}{|X_{bu}\wedge X_{bv}|}(q_{b})=-N(p), +\] + +\end_inset + +luego +\begin_inset Formula $N(p)=0\#$ +\end_inset + +. + Por tanto +\begin_inset Formula $\det(Jh)>0$ +\end_inset + + y las cartas del atlas +\begin_inset Formula $\{(U_{p},X_{p})\}_{p\in S}$ +\end_inset + + son compatibles. +\end_layout + +\begin_layout Standard +En adelante, cuando consideremos una parametrización +\begin_inset Formula $(U,X)$ +\end_inset + +, escribiremos +\begin_inset Formula $N(u,v):=N(X(u,v))$ +\end_inset + +, +\begin_inset Formula $N_{u}:=\frac{\partial(N\circ X)}{\partial u}$ +\end_inset + + y +\begin_inset Formula $N_{v}:=\frac{\partial(N\circ X)}{\partial v}$ +\end_inset + +. + En general, para +\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $f_{x_{i}}:=\frac{\partial f}{\partial x_{i}}$ +\end_inset + +. +\end_layout + +\begin_layout Section +La segunda forma fundamental +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie orientada con aplicación de Gauss +\begin_inset Formula $N:S\to\mathbb{S}^{2}$ +\end_inset + +, llamamos +\series bold +imagen esférica +\series default + de +\begin_inset Formula $S$ +\end_inset + + a +\begin_inset Formula $\text{Im}N\subseteq\mathbb{S}^{2}$ +\end_inset + +. + Ejemplos: +\end_layout + +\begin_layout Enumerate +La imagen esférica de un plano es unipuntual. +\end_layout + +\begin_deeper +\begin_layout Standard +Dado el plano +\begin_inset Formula $\Pi:=p_{0}+\langle v\rangle\subseteq\mathbb{R}^{3}$ +\end_inset + +, donde podemos suponer +\begin_inset Formula $v$ +\end_inset + + unitario, la imagen de +\begin_inset Formula $N(p):=v$ +\end_inset + + es +\begin_inset Formula $\{v\}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La imagen esférica de +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + + es +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +La aplicación de Gauss es +\begin_inset Formula $\pm1_{\mathbb{S}^{2}}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La imagen esférica de un grafo +\begin_inset Formula $\{(x,y,f(x,y))\}_{(x,y)\in U}$ +\end_inset + + con +\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + + diferenciable está contenida en el hemisferio (estricto) norte o sur. +\end_layout + +\begin_deeper +\begin_layout Standard +Una orientación es +\begin_inset Formula $N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}(u,v)$ +\end_inset + +, y como la coordenada +\begin_inset Formula $z$ +\end_inset + + de +\begin_inset Formula $N$ +\end_inset + + es siempre positiva, +\begin_inset Formula $\text{Im}N$ +\end_inset + + está en el hemisferio norte estricto. + Con la orientación opuesta está en el hemisferio sur estricto. +\end_layout + +\end_deeper +\begin_layout Enumerate +La imagen esférica de un cilindro es un circulo máximo de la esfera. +\end_layout + +\begin_deeper +\begin_layout Standard +Los cilindros se obtienen por un movimiento de +\begin_inset Formula $S_{r}:=\{x^{2}+y^{2}=r^{2}\}$ +\end_inset + + para algún +\begin_inset Formula $r>0$ +\end_inset + +, y como su orientación es +\begin_inset Formula $N(x,y,z)=\pm\frac{1}{r}(x,y,0)$ +\end_inset + +, +\begin_inset Formula $N(S_{r})=\{\frac{1}{r}(x,y,0):x^{2}+y^{2}=r^{2}\}=\{(x,y,0):x^{2}+y^{2}=1\}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +El +\series bold +catenoide +\series default +, +\begin_inset Formula $C:=\{x^{2}+y^{2}=\cosh^{2}z\}$ +\end_inset + +, tiene imagen esférica +\begin_inset Formula $\mathbb{S}^{2}\setminus\{\mathsf{N},\mathsf{S}\}$ +\end_inset + +, donde +\begin_inset Formula $\mathsf{N}:=(0,0,1)$ +\end_inset + + es el +\series bold +polo norte +\series default + y +\begin_inset Formula $\mathsf{S}:=(0,0,-1)$ +\end_inset + + es el +\series bold +polo sur +\series default +. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}-\cosh^{2}z$ +\end_inset + +, como +\begin_inset Formula $f_{x}=2x$ +\end_inset + +, +\begin_inset Formula $f_{y}=2y$ +\end_inset + + y +\begin_inset Formula $f_{z}=-2\cosh z\sinh z$ +\end_inset + +, el único punto crítico de +\begin_inset Formula $f$ +\end_inset + + es el origen, con +\begin_inset Formula $f(0)=-1$ +\end_inset + +, de modo que 0 es un valor regular de +\begin_inset Formula $f\in{\cal C}^{\infty}$ +\end_inset + + y +\begin_inset Formula $C=\{f(x,y,z)=0\}$ +\end_inset + + es una superficie de nivel regular y +\begin_inset Formula +\begin{align*} +N(x,y,z) & =\frac{\nabla f(x,y,z)}{\Vert\nabla f(x,y,z)\Vert}=\frac{(2x,2y,-2\cosh z\sinh z)}{2\sqrt{x^{2}+y^{2}+\cosh^{2}z\sinh^{2}z}}\\ + & =\frac{(x,y,-\cosh z\sinh z)}{\sqrt{\cosh^{2}z+\cosh^{2}z\sinh^{2}z}}=\frac{(x,y,-\cosh z\sinh z)}{\cosh^{2}z}. +\end{align*} + +\end_inset + +Como +\begin_inset Formula $N_{1}(p)^{2}+N_{2}(p)^{2}=\frac{x^{2}+y^{2}}{\cosh^{4}z}=\frac{1}{\cosh^{2}z}>0$ +\end_inset + +, no se cubren los polos norte y sur. + Sean ahora +\begin_inset Formula $(\hat{x},\hat{y},\hat{z})\in\mathbb{S}^{2}\setminus\{\mathsf{N},\mathsf{S}\}$ +\end_inset + +, +\begin_inset Formula $z:=\arg\tanh(-\hat{z})$ +\end_inset + + (que existe porque +\begin_inset Formula $\hat{z}\in(-1,1)$ +\end_inset + +), +\begin_inset Formula $x:=\hat{x}\cosh^{2}z$ +\end_inset + + e +\begin_inset Formula $y:=\hat{y}\cosh^{2}z$ +\end_inset + +, es claro que +\begin_inset Formula $N(x,y,z)=(\hat{x},\hat{y},\hat{z})$ +\end_inset + +. + Ahora bien, +\begin_inset Formula +\begin{multline*} +x^{2}+y^{2}=(\hat{x}^{2}+\hat{y}^{2})\cosh^{4}z=(1-\hat{z}^{2})\cosh^{4}z=\left(1-\tanh^{2}z\right)\cosh^{4}z=\\ +=\frac{\cosh^{2}z-\sinh^{2}z}{\cosh^{2}z}\cosh^{4}z=\frac{\cosh^{4}z}{\cosh^{2}z}=\cosh^{2}z, +\end{multline*} + +\end_inset + +luego +\begin_inset Formula $(x,y,z)\in C$ +\end_inset + + y +\begin_inset Formula $N(x,y,z)$ +\end_inset + + cubre +\begin_inset Formula $\mathbb{S}^{2}\setminus\{\mathsf{N},\mathsf{S}\}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Para +\begin_inset Formula $p\in\mathbb{S}^{2}$ +\end_inset + + es +\begin_inset Formula $T_{N(p)}\mathbb{S}^{2}=T_{p}\mathbb{S}^{2}$ +\end_inset + +, pues +\begin_inset Formula $N(p)=\pm p$ +\end_inset + + y +\begin_inset Formula $T_{-p}\mathbb{S}^{2}=\langle N(-p)\rangle^{\bot}=\langle p\rangle^{\bot}=\langle N(p)\rangle^{\bot}=T_{p}\mathbb{S}^{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada por +\begin_inset Formula $N$ +\end_inset + +, llamamos +\series bold +operador forma +\series default + o +\series bold +endomorfismo de Weingarten +\series default + en +\begin_inset Formula $p\in S$ +\end_inset + + a +\begin_inset Formula $A_{p}:=-dN_{p}:T_{p}S\to T_{p}S$ +\end_inset + +. + En efecto, como +\begin_inset Formula $N:S\to\mathbb{S}^{2}$ +\end_inset + +, +\begin_inset Formula $dN_{p}:T_{p}S\to T_{N(p)}\mathbb{S}^{2}$ +\end_inset + +, pero como la normal en +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + + es +\begin_inset Formula $1_{\mathbb{S}^{2}}$ +\end_inset + +, +\begin_inset Formula $T_{p'}\mathbb{S}^{2}=\langle p'\rangle^{\bot}$ +\end_inset + + para todo +\begin_inset Formula $p'\in\mathbb{S}^{2}$ +\end_inset + + y en particular +\begin_inset Formula $T_{N(p)}\mathbb{S}^{2}=\langle N(p)\rangle^{\bot}=T_{p}S$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $A_{p}$ +\end_inset + + es +\series bold +autoadjunto +\series default +, es decir, +\begin_inset Formula $\langle A_{p}v,w\rangle=\langle v,A_{p}w\rangle$ +\end_inset + +. + +\series bold +Demostración: +\series default + Por linealidad, basta demostrarlo para una base de +\begin_inset Formula $T_{p}S$ +\end_inset + +. + Sean +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q:=(u_{0},v_{0}):=X^{-1}(p)$ +\end_inset + +, tomamos la base +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ +\end_inset + + y queremos ver que +\begin_inset Formula $\langle dN_{p}(X_{u}(q)),X_{v}(q)\rangle=\langle X_{u}(q),dN_{p}(X_{v}(q))\rangle$ +\end_inset + +. + Sea entonces +\begin_inset Formula $\alpha(u):=X(u_{0}+u,v_{0})$ +\end_inset + +, +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=X_{u}(q)$ +\end_inset + +, luego +\begin_inset Formula $dN_{p}(X_{u}(q))=\frac{\partial(N\circ\alpha)}{\partial u}(0)=\frac{\partial(N\circ X)}{\partial u}(u_{0},v_{0})=N_{u}(u_{0},v_{0})$ +\end_inset + +. + Análogamente +\begin_inset Formula $dN_{p}(X_{v}(q))=N_{v}(u_{0},v_{0})$ +\end_inset + +, por lo que queda ver que +\begin_inset Formula $\langle N_{u},X_{v}\rangle(q)=\langle N_{v},X_{u}\rangle(q)$ +\end_inset + +. + Sabemos que +\begin_inset Formula $\langle N,X_{u}\rangle=\langle N,X_{v}\rangle=0$ +\end_inset + +, y derivando, +\begin_inset Formula $\langle N_{v},X_{u}\rangle+\langle N,X_{uv}\rangle=\langle N_{u},X_{v}\rangle+\langle N,X_{vu}\rangle=0$ +\end_inset + +, pero +\begin_inset Formula $X_{uv}=X_{vu}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Ejemplos: +\end_layout + +\begin_layout Enumerate +Para un plano, +\begin_inset Formula $A_{p}\equiv0$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $N$ +\end_inset + + es fijo, luego +\begin_inset Formula $-dN_{p}\equiv0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Para +\begin_inset Formula $\mathbb{S}^{2}(r)$ +\end_inset + + orientada con +\begin_inset Formula $N(p)=\pm\frac{1}{r}p$ +\end_inset + +, +\begin_inset Formula $A_{p}=\mp\frac{1}{r}1_{T_{p}\mathbb{S}^{2}(r)}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Para el cilindro +\begin_inset Formula $X(\mathbb{R}^{2})$ +\end_inset + + con +\begin_inset Formula $X(u,v):=(r\cos u,r\sin u,v)$ +\end_inset + +, si +\begin_inset Formula $p\in C$ +\end_inset + + y +\begin_inset Formula $q\in X^{-1}(p)$ +\end_inset + +, +\begin_inset Formula $A_{p}=\text{diag}(-\frac{1}{r},0)$ +\end_inset + + respecto a la base +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $p=:(x,y,z)$ +\end_inset + + y +\begin_inset Formula $q=:(u,v)$ +\end_inset + +, +\begin_inset Formula $X_{u}(q)=(-r\sin u,r\cos u,0)$ +\end_inset + +, +\begin_inset Formula $X_{v}(q)=(0,0,1)$ +\end_inset + + y, como +\begin_inset Formula $N(x,y,z)=\frac{1}{r}(x,y,0)=(\cos u,\sin u,0)$ +\end_inset + +, +\begin_inset Formula $N_{u}(q)=(-\sin u,\cos u,0)=-\frac{1}{r}X_{u}$ +\end_inset + + y +\begin_inset Formula $N_{v}(q)=0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Para el +\series bold +paraboloide hiperbólico +\series default + o +\series bold +silla de montar +\series default +, +\begin_inset Formula $S:=\{y^{2}-x^{2}=z\}=\{(u,v,v^{2}-u^{2})\}_{(u,v)\in\mathbb{R}^{2}}$ +\end_inset + +, +\begin_inset Formula $A_{p}(0)\equiv\text{diag}(-2,2)$ +\end_inset + + respecto a la base +\begin_inset Formula $(X_{u}(0),X_{v}(0))$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $S$ +\end_inset + + es una superficie porque es el grafo de +\begin_inset Formula $f:\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula $f(u,v):=v^{2}-u^{2}$ +\end_inset + +. + Entonces +\begin_inset Formula +\[ +N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}=\frac{(2u,-2v,1)}{\sqrt{1+4u^{2}+4v^{2}}}, +\] + +\end_inset + +luego +\begin_inset Formula +\begin{align*} +N_{u}(u,v) & =\frac{(2(1+4u^{2}+4v^{2})-8u^{2},8uv,-4u)}{(1+4u^{2}+4v^{2})^{3/2}}=\frac{(2(1+4v^{2}),8uv,-4u)}{(1+4u^{2}+4v^{2})^{3/2}},\\ +N_{v}(u,v) & =\frac{(-8uv,-2(1+4u^{2}+4v^{2})+8v^{2},-4v)}{(1+4u^{2}+4v^{2})^{3/2}}=\frac{(-8uv,-2(1+4u^{2}),-4v)}{(1+4u^{2}+4v^{2})^{3/2}}, +\end{align*} + +\end_inset + +y en particular +\begin_inset Formula $N_{u}(0)=(2,0,0)$ +\end_inset + + y +\begin_inset Formula $N_{v}(0)=(0,-2,0)$ +\end_inset + +, pero +\begin_inset Formula $X_{u}(0)=(1,0,0)$ +\end_inset + + y +\begin_inset Formula $X_{v}(0)=(0,1,0)$ +\end_inset + +, luego +\begin_inset Formula $N_{u}(0)=2X_{u}(0)$ +\end_inset + + y +\begin_inset Formula $N_{v}(0)=2X_{v}(0)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +El operador forma +\begin_inset Formula $A_{p}$ +\end_inset + + lleva asociada unívocamente una forma bilineal simétrica +\begin_inset Formula $\sigma_{p}:T_{p}S\times T_{p}S\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula $\sigma_{p}(v,w):=\langle A_{p}v,w\rangle$ +\end_inset + +, así como una forma cuadrática +\begin_inset Formula ${\cal II}_{p}:T_{p}S\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula ${\cal II}_{p}(v):=\sigma_{p}(v,v)=\langle A_{p}v,v\rangle$ +\end_inset + +. + +\begin_inset Formula ${\cal II}_{p}$ +\end_inset + + es la +\series bold +segunda forma fundamental +\series default + de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Las tres formas dan la misma información usando la +\series bold +identidad de polarización: +\series default + +\begin_inset Formula +\[ +\sigma_{p}(v,w)=\frac{1}{2}\left({\cal II}_{p}(v+w)-{\cal II}_{p}(v)-{\cal II}_{p}(w)\right). +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Curvas geodésica y normal +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular y +\begin_inset Formula $V:\mathbb{R}\to T_{p}S$ +\end_inset + + diferenciable, llamamos +\series bold +derivada covariante +\series default + a +\begin_inset Formula +\[ +\frac{DV}{dt}(t):=\pi_{T_{p}S}V'(t), +\] + +\end_inset + +la proyección de +\begin_inset Formula $V'(t)$ +\end_inset + + en +\begin_inset Formula $T_{p}S$ +\end_inset + +. + Propiedades: Sean +\begin_inset Formula $V,W:\mathbb{R}\to T_{p}S$ +\end_inset + + y +\begin_inset Formula $f:I\subseteq\mathbb{R}\to\mathbb{R}$ +\end_inset + + diferenciables, siendo +\begin_inset Formula $I$ +\end_inset + + un intervalo: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\frac{D(fV)}{dt}=f'V+f\frac{DV}{dt}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $\pi:=\pi_{T_{p}S}$ +\end_inset + +, +\begin_inset Formula $\frac{D(fV)}{dt}=\pi((fV)')=\pi(fV'+f'V)=f\pi V'+f'\pi V=f\frac{DV}{dt}+f'V$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\frac{D(V+W)}{dt}=\frac{DV}{dt}+\frac{DW}{dt}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $\frac{D(V+W)}{dt}=\pi((V+W)')=\pi V'+\pi W'=\frac{DV}{dt}+\frac{DW}{dt}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{DV}{dt}W\rangle+\langle V,\frac{DW}{dt}\rangle$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{dV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle$ +\end_inset + +, pero dada una base ortonormal +\begin_inset Formula $(v_{1},v_{2},v_{3})$ +\end_inset + + con +\begin_inset Formula $T_{p}S=\text{span}\{v_{1},v_{2}\}$ +\end_inset + +, si +\begin_inset Formula $\frac{dV}{dt}(t)=\sum_{i}x_{i}v_{i}$ +\end_inset + + y +\begin_inset Formula $W(t)=\sum_{i}y_{i}v_{i}$ +\end_inset + +, +\begin_inset Formula $\langle\frac{dV}{dt}(t),W(t)\rangle=\sum_{i=1}^{3}x_{i}y_{i}\overset{y_{3}=0}{=}x_{1}y_{1}+x_{2}y_{2}=\langle\pi_{T_{p}S}\frac{dV}{dt}(t),W(t)\rangle=\langle\frac{DV}{dt}(t),W(t)\rangle$ +\end_inset + +, y análogamente para +\begin_inset Formula $\langle V,\frac{dW}{dt}\rangle$ +\end_inset + +, luego +\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{DV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada por +\begin_inset Formula $N$ +\end_inset + + y +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva, entonces +\begin_inset Formula $\alpha'(t)\in T_{\alpha(t)}S$ +\end_inset + + para +\begin_inset Formula $t\in I$ +\end_inset + +, pero en general +\begin_inset Formula $\alpha''(t)\notin T_{\alpha(t)}S$ +\end_inset + +, aunque se escribe de forma única como la suma de una +\series bold +aceleración tangencial +\series default + o +\series bold +intrínseca +\series default + +\begin_inset Formula $\alpha''(t)^{\top}\in T_{\alpha(t)}S$ +\end_inset + + y una +\series bold +aceleración normal +\series default + o +\series bold +extrínseca +\series default + +\begin_inset Formula $\alpha''(t)^{\bot}\in\text{span}\{N(\alpha(t))\}$ +\end_inset + +. + Como +\begin_inset Formula $\alpha''(t)^{\top}=\frac{D\alpha'}{dt}$ +\end_inset + +, +\begin_inset Formula +\[ +\alpha''(t)=\frac{D\alpha'}{dt}(t)+\langle\alpha''(t),N(\alpha(t))\rangle N(\alpha(t)). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva parametrizada por longitud de arco, el +\series bold +triedro de Darboux +\series default + es la base ortonormal positivamente orientada +\begin_inset Formula $(\alpha'(s),J\alpha'(s):=\alpha'(s)\wedge N(\alpha(s)),N(\alpha(s))\rangle$ +\end_inset + +. + Entonces +\begin_inset Formula $\frac{D\alpha'}{ds}(s)=\kappa_{g}(s)J\alpha'(s)$ +\end_inset + +, donde +\begin_inset Formula $\kappa_{g}:=\langle\alpha'',J\alpha'\rangle:I\to\mathbb{R}$ +\end_inset + +, es la +\series bold +curvatura geodésica +\series default + de +\begin_inset Formula $\alpha$ +\end_inset + +, cuyo signo depende de +\begin_inset Formula $N$ +\end_inset + +. + En efecto, +\begin_inset Formula +\begin{multline*} +\langle\frac{D\alpha'}{ds}(s),\alpha'(s)\rangle=\langle\alpha''(s)-\langle\alpha''(s),N(\alpha(s))\rangle N(\alpha(s)),\alpha'(s)\rangle=\\ +=\langle\alpha''(s),\alpha'(s)\rangle-\langle\alpha''(s),N(\alpha(s))\rangle\langle N(\alpha(s)),\alpha'(s)\rangle=0, +\end{multline*} + +\end_inset + +y +\begin_inset Formula $\kappa_{g}(s)=\langle\frac{D\alpha'}{ds}(s),J\alpha'(s)\rangle=\langle\alpha''(s),J\alpha'(s)\rangle$ +\end_inset + +, pero +\begin_inset Formula $J\alpha'(s)$ +\end_inset + + puede ser un vector o su opuesto según lo sea +\begin_inset Formula $N$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Dada una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + +, +\begin_inset Formula ${\cal II}_{\alpha(t)}(\alpha'(t))=\langle\alpha''(t),N(\alpha(t))\rangle$ +\end_inset + +. + En efecto, como +\begin_inset Formula $\alpha'(t)\in T_{\alpha(t)}S$ +\end_inset + + para cada +\begin_inset Formula $t$ +\end_inset + +, +\begin_inset Formula $\langle\alpha'(t),N(\alpha(t))\rangle=0$ +\end_inset + + y, derivando, +\begin_inset Formula $\langle\alpha''(t),N(\alpha(t))\rangle+\langle\alpha'(t),(N\circ\alpha)'(t)\rangle=0$ +\end_inset + +, pero +\begin_inset Formula $(N\circ\alpha)'(t)=dN_{\alpha(t)}(\alpha'(t))$ +\end_inset + +, luego +\begin_inset Formula $\langle\alpha''(t),N(\alpha(t))\rangle=-\langle\alpha'(t),dN_{\alpha(t)}(\alpha'(t))\rangle=\langle\alpha'(t),A_{\alpha(t)}\alpha'(t)\rangle={\cal II}_{\alpha(t)}(\alpha'(t))$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +Entonces, dados +\begin_inset Formula $p\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + unitario, llamamos +\series bold +curvatura normal +\series default + de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + en la dirección de +\begin_inset Formula $v$ +\end_inset + + a +\begin_inset Formula $\kappa_{n}(v,p):={\cal II}_{p}(v)=\langle\alpha''(0),N(p)\rangle$ +\end_inset + +, siendo +\begin_inset Formula $\alpha:(-\delta,\delta)\to S$ +\end_inset + + una curva con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Ejemplos: +\end_layout + +\begin_layout Enumerate +Un plano tiene curvatura normal 0 en todo punto y dirección. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $A_{p}=0$ +\end_inset + +, +\begin_inset Formula $\kappa_{n}(v,p)={\cal II}_{p}(v)=\langle A_{p}v,v\rangle=0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\mathbb{S}^{2}(r)$ +\end_inset + + tiene curvatura normal constante +\begin_inset Formula $-\frac{1}{r}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $N(p)=\frac{1}{r}p$ +\end_inset + +, +\begin_inset Formula $\kappa_{n}(v,p)=\langle A_{p}v,v\rangle=\langle-\frac{1}{r}v,v\rangle=-\frac{1}{r}|v|^{2}=-\frac{1}{r}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Dados +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + unitario y +\begin_inset Formula $\Pi_{v}:=\text{span}\{v,N(p)\}$ +\end_inset + +, llamamos +\series bold +sección normal +\series default + +\begin_inset Formula $C_{v}$ +\end_inset + + a la curva regular plana resultante de intersecar +\begin_inset Formula $S$ +\end_inset + + con +\begin_inset Formula $\Pi_{v}$ +\end_inset + +. + Sea entonces +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una parametrización por arco de +\begin_inset Formula $C_{v}$ +\end_inset + + con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +, entonces +\begin_inset Formula $\kappa_{n}(v,p)=\kappa(0)$ +\end_inset + +, siendo +\begin_inset Formula $\kappa$ +\end_inset + + la curvatura de +\begin_inset Formula $\alpha$ +\end_inset + + como curva plana. + En efecto, como +\begin_inset Formula $v\in T_{p}S$ +\end_inset + +, +\begin_inset Formula $v\bot N(p)$ +\end_inset + + y el vector normal es +\begin_inset Formula $\mathbf{n}=J_{\Pi_{v}}v=\pm N(p)$ +\end_inset + +, y como todavía no hemos orientado el plano podemos tomar +\begin_inset Formula $\mathbf{n}=N(p)$ +\end_inset + +, pero entonces +\begin_inset Formula $\kappa_{n}(v,p)=\langle\alpha''(0),N(p)\rangle=\langle\kappa(0)\mathbf{n}(0),N(p)\rangle=\kappa(0)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + es una curva parametrizada por arco, +\begin_inset Formula $\alpha''(s)=\kappa_{g}(s)J\alpha'(s)+\kappa_{n}(s)N(\alpha(s))$ +\end_inset + +, siendo +\begin_inset Formula $\kappa_{n}(s):=\kappa_{n}(\alpha'(s),\alpha(s))=\langle\alpha''(s),N(\alpha(s))\rangle$ +\end_inset + +, luego +\begin_inset Formula +\[ +\kappa(s)^{2}=\kappa_{g}(s)^{2}+\kappa_{n}(s)^{2}. +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Curvaturas principales +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{AAlG} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Toda matriz simétrica real +\begin_inset Formula $A\in{\cal M}_{m}(\mathbb{R})$ +\end_inset + + admite una matriz ortogonal +\begin_inset Formula $P$ +\end_inset + + tal que +\begin_inset Formula $P^{-1}AP=P^{t}AP$ +\end_inset + + es diagonal. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dados una superficie regular +\begin_inset Formula $S$ +\end_inset + + orientada y +\begin_inset Formula $p\in S$ +\end_inset + +, existe una base ortonormal +\begin_inset Formula $(e_{1},e_{2})$ +\end_inset + + en la que +\begin_inset Formula $A_{p}$ +\end_inset + + es diagonal, pues +\begin_inset Formula $A_{p}$ +\end_inset + + es simétrica. + Si +\begin_inset Formula $\kappa_{1}(p)$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)$ +\end_inset + + son los valores propios asociados respectivamente a +\begin_inset Formula $e_{1}$ +\end_inset + + y +\begin_inset Formula $e_{2}$ +\end_inset + +, podemos suponer que +\begin_inset Formula $\kappa_{1}(p)\leq\kappa_{2}(p)$ +\end_inset + +, y llamamos +\series bold +curvaturas principales +\series default + de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + a +\begin_inset Formula $\kappa_{1}(p)$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)$ +\end_inset + + y +\series bold +direcciones principales +\series default + a +\begin_inset Formula $e_{1}$ +\end_inset + + y +\begin_inset Formula $e_{2}$ +\end_inset + +, o a todos los vectores unitarios de +\begin_inset Formula $T_{p}S$ +\end_inset + + si +\begin_inset Formula $\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +, pues en tal caso todos los vectores no nulos son propios al ser +\begin_inset Formula $A_{p}$ +\end_inset + + una homotecia. + Se tiene +\begin_inset Formula $\kappa_{1}(p)=\kappa_{n}(e_{1},p)$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)=\kappa_{n}(e_{2},p)$ +\end_inset + +, pues +\begin_inset Formula $\kappa_{n}(e_{i},p)=\langle A_{p}e_{i},e_{i}\rangle=\langle\kappa_{i}(p)e_{i},e_{i}\rangle=\kappa_{i}(p)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Ejemplos: +\end_layout + +\begin_layout Enumerate +Todas las direcciones del plano y la esfera son principales. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $\kappa_{n}$ +\end_inset + + es constante, +\begin_inset Formula $\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +El cilindro +\begin_inset Formula $\{x^{2}+y^{2}=r^{2}\}$ +\end_inset + + tiene como curvaturas principales +\begin_inset Formula $-\frac{1}{r}$ +\end_inset + + y 0. +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $C:=\{x^{2}+y^{2}=r^{2}\}=\{X(u,v):=(r\cos u,r\sin u,v)\}_{u,v\in\mathbb{R}}$ +\end_inset + +, +\begin_inset Formula $p=(x,y,z)\in C$ +\end_inset + + y la orientación +\begin_inset Formula $N(p):=\frac{1}{r}(x,y,0)$ +\end_inset + +, entonces +\begin_inset Formula $X_{u}=(-r\sin u,r\cos u,0)$ +\end_inset + +, +\begin_inset Formula $X_{v}=e_{3}$ +\end_inset + + y +\begin_inset Formula $N(u,v)=(\cos u,\sin u,0)$ +\end_inset + +, luego +\begin_inset Formula $A_{p}=-(-\sin u,\cos u,0)=-\frac{1}{r}X_{u}$ +\end_inset + + y por tanto +\begin_inset Formula $A_{p}\equiv\text{diag}(-\frac{1}{r},0)$ +\end_inset + + con la base +\begin_inset Formula $(X_{u},X_{v})$ +\end_inset + + de +\begin_inset Formula $T_{p}S$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La silla de montar tiene curvaturas principales +\begin_inset Formula $-2$ +\end_inset + + y 2 en el origen. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $A_{p}\equiv\text{diag}(-2,2)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Una +\series bold +línea de curvatura +\series default + en una superficie regular orientada +\begin_inset Formula $S$ +\end_inset + + es una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + tal que +\begin_inset Formula $\alpha'(t)$ +\end_inset + + es una dirección principal de +\begin_inset Formula $\alpha(t)$ +\end_inset + + para todo +\begin_inset Formula $t\in I$ +\end_inset + +. + Si las curvaturas principales son distintas en todo punto de un abierto + +\begin_inset Formula $V\subseteq S$ +\end_inset + +, por cada +\begin_inset Formula $p\in V$ +\end_inset + + pasan dos únicas líneas de curvatura y estas se cortan de forma ortogonal. +\end_layout + +\begin_layout Standard + +\series bold +Fórmula de Euler: +\series default + Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $\kappa_{1}(p)\leq\kappa_{2}(p)$ +\end_inset + + las curvaturas principales de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +, +\begin_inset Formula $e_{1}$ +\end_inset + + y +\begin_inset Formula $e_{2}$ +\end_inset + + las respectivas direcciones principales, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + y +\begin_inset Formula $\theta$ +\end_inset + + tal que +\begin_inset Formula $\cos\theta=\langle e_{1},v\rangle$ +\end_inset + +, entonces +\begin_inset Formula $\kappa_{n}(v,p)=\kappa_{1}(p)\cos^{2}\theta+\kappa_{2}(p)\sin^{2}\theta$ +\end_inset + +. + En efecto, sea +\begin_inset Formula $v=:\cos\omega e_{1}+\sin\omega e_{2}$ +\end_inset + +, +\begin_inset Formula $\kappa_{n}(v,p)=\langle A_{p}v,v\rangle=\langle\kappa_{1}(p)\cos\omega e_{1}+\kappa_{2}(p)\cos\omega e_{2},\cos\omega e_{1}+\sin\omega e_{2}\rangle=\kappa_{1}(p)\cos^{2}\omega+\kappa_{2}(p)\sin^{2}\omega$ +\end_inset + +, y aunque +\begin_inset Formula $\omega=\pm\theta+2k\pi$ +\end_inset + + para algún +\begin_inset Formula $k\in\mathbb{Z}$ +\end_inset + +, el coseno y por tanto el cuadrado del seno coinciden. +\end_layout + +\begin_layout Standard +Con esto, +\begin_inset Formula $\kappa_{1}(p)=\min\{\kappa_{n}(v,p)\}_{|v|=1}$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)=\max\{\kappa_{n}(v,p)\}_{|v|=1}$ +\end_inset + +, pues por la fórmula, si +\begin_inset Formula $|v|=1$ +\end_inset + +, +\begin_inset Formula $\kappa_{n}(v,p)=\kappa_{1}(p)(1-\sin^{2}\theta)+\kappa_{2}(p)\sin^{2}\theta$ +\end_inset + + para algún +\begin_inset Formula $\theta$ +\end_inset + +. + Llamamos +\series bold +curvatura mínima +\series default + a +\begin_inset Formula $\kappa_{1}(p)$ +\end_inset + + y +\series bold +curvatura máxima +\series default + a +\begin_inset Formula $\kappa_{2}(p)$ +\end_inset + +. + La +\series bold +curvatura de Gauss +\series default + de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p\in S$ +\end_inset + + es +\begin_inset Formula $K(p):=\det A_{p}=\kappa_{1}(p)\kappa_{2}(p)$ +\end_inset + +, y la +\series bold +curvatura media +\series default + es +\begin_inset Formula $H(p):=\frac{1}{2}\text{tr}A_{p}=\frac{1}{2}(\kappa_{1}(p)+\kappa_{2}(p))$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Las curvaturas máxima, mínima y media cambian de signo al cambiar de orientación. + La curvatura de Gauss no, pues es el producto de dos curvaturas que cambian + de signo a la vez. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie regular, +\begin_inset Formula $p\in S$ +\end_inset + + es +\series bold +elíptico +\series default + si +\begin_inset Formula $K(p)>0$ +\end_inset + +, +\series bold +hiperbólico +\series default + si +\begin_inset Formula $K(p)<0$ +\end_inset + +, +\series bold +parabólico +\series default + si +\begin_inset Formula $K(p)=0$ +\end_inset + + pero +\begin_inset Formula $A_{p}\not\equiv0$ +\end_inset + + y +\series bold +llano +\series default + o +\series bold +plano +\series default + si +\begin_inset Formula $A_{p}\equiv0$ +\end_inset + +. + Ejemplos: +\end_layout + +\begin_layout Enumerate +Los puntos de un plano son planos. +\end_layout + +\begin_layout Enumerate +Los puntos de una esfera son elípticos. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $r$ +\end_inset + + es el radio, +\begin_inset Formula $K(p)=(-\frac{1}{r})^{2}=\frac{1}{r^{2}}>0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +El origen en la silla de montar es hiperbólico. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $A_{p}\equiv\text{diag}(-2,2)$ +\end_inset + + respecto de cierta base, luego +\begin_inset Formula $K(p)=-4$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Los puntos de un cilindro son parabólicos. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $r$ +\end_inset + + es el radio, +\begin_inset Formula $A_{p}\equiv\text{diag}(-\frac{1}{r},0)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +En +\begin_inset Formula $\{z=(x^{2}+y^{2})^{2}\}$ +\end_inset + +, el origen es un punto plano. +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +La superficie es el grafo +\begin_inset Formula $S:=\{X(u,v):=(u,v,(u^{2}+v^{2})^{2}\}_{u,v\in\mathbb{R}}$ +\end_inset + +, de modo que +\begin_inset Formula $X_{u}=(1,0,2(u^{2}+v^{2})u)$ +\end_inset + +, +\begin_inset Formula $X_{v}=(0,1,2(u^{2}+v^{2})v)$ +\end_inset + +, +\begin_inset Formula $N=\frac{(-4(u^{2}+v^{2})u,-4(u^{2}+v^{2})v,1)}{\sqrt{16(u^{2}+v^{2})^{2}+1}}$ +\end_inset + +, +\begin_inset Formula $N_{u}=\frac{(-4(3u^{2}+v^{2})(16(u^{2}+v^{2})^{2}+1)+256(u^{2}+v^{2})^{2}u^{2},-8uv(16(u^{2}+v^{2})^{2}+1)+256(u^{2}+v^{2})^{2}uv,64(u^{2}+v^{2})u)}{(16(u^{2}+v^{2})^{2}+1)^{3/2}}$ +\end_inset + + y entonces +\begin_inset Formula $N_{u}(0,0)=(0,0,0)$ +\end_inset + + y, por simetría, +\begin_inset Formula $N_{v}(0,0)=(0,0,0)$ +\end_inset + +, por lo que +\begin_inset Formula $A_{p}\equiv0$ +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +En una superficie regular +\begin_inset Formula $S$ +\end_inset + + orientada, +\begin_inset Formula $p\in S$ +\end_inset + + es un +\series bold +punto umbilical +\series default + si +\begin_inset Formula $\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +. + +\begin_inset Formula $S$ +\end_inset + + es +\series bold +totalmente umbilical +\series default + si todos sus puntos son umbilicales. + Así, el plano y la esfera son totalmente umbilicales. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, toda superficie regular, orientable con orientación +\begin_inset Formula ${\cal C}^{2}$ +\end_inset + +, conexa y totalmente umbilical es un trozo de esfera o plano. +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula $S$ +\end_inset + + la superficie y +\begin_inset Formula $N$ +\end_inset + + una orientación de +\begin_inset Formula $S$ +\end_inset + +, para +\begin_inset Formula $p\in S$ +\end_inset + + es +\begin_inset Formula $H(p)=\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +, luego +\begin_inset Formula $A_{p}\equiv\text{diag}(H(p),H(p))$ +\end_inset + + y +\begin_inset Formula $A_{p}=H(p)1_{T_{p}S}$ +\end_inset + +. + +\begin_inset Formula $H:S\to\mathbb{R}$ +\end_inset + + es diferenciable, y queremos ver que es constante. + Sean +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +, +\begin_inset Formula $q:=(u_{0},v_{0}):=X^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $\alpha(u):=X(u_{0}+u,v_{0})$ +\end_inset + +, como +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=q$ +\end_inset + +, +\begin_inset Formula $dH_{p}(X_{u}(q))=\frac{d(H\circ\alpha)}{dt}(0)=\frac{d}{dt}(H(X(u_{0}+u,v_{0})))(0)=(H\circ X)_{u}(q)$ +\end_inset + +, y por simetría +\begin_inset Formula $dH_{p}(X_{v}(q))=\frac{\partial(H\circ X)}{\partial v}(q)$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +Como +\begin_inset Formula $A_{p}=H(p)1_{T_{p}S}$ +\end_inset + +, +\begin_inset Formula $(H\circ X)(q)X_{u}(q)=H(p)X_{u}(q)=A_{p}(X_{u}(q))=-dN_{p}(X_{u}(q))=-(N\circ X)(q)$ +\end_inset + +, y como esto es cierto para todo +\begin_inset Formula $q\in U$ +\end_inset + +, +\begin_inset Formula $(N\circ X)_{u}=-(H\circ X)X_{u}$ +\end_inset + +, y por simetría +\begin_inset Formula $(N\circ X)_{v}=-(H\circ X)X_{v}$ +\end_inset + +. + Derivando, +\begin_inset Formula $(N\circ X)_{uv}=-(H\circ X)_{v}X_{u}-(H\circ X)X_{uv}$ +\end_inset + + y +\begin_inset Formula $(N\circ X)_{vu}=-(H\circ X)_{u}X_{v}-(H\circ X)X_{vu}$ +\end_inset + +, y como las derivadas cruzadas coinciden, +\begin_inset Formula $(H\circ X)_{v}X_{u}=(H\circ X)_{u}X_{v}$ +\end_inset + +. + Como +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ +\end_inset + + es una base en cada +\begin_inset Formula $q\in U$ +\end_inset + +, necesariamente +\begin_inset Formula $(H\circ X)_{u},(H\circ X)_{v}\equiv0$ +\end_inset + +, luego +\begin_inset Formula $dH_{p}(X_{u}(q)),dH_{p}(X_{v}(q))=0$ +\end_inset + + y al ser +\begin_inset Formula $S$ +\end_inset + + conexa, +\begin_inset Formula $H\equiv c$ +\end_inset + + para algún +\begin_inset Formula $c\in\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $c=0$ +\end_inset + +, +\begin_inset Formula $H\equiv0$ +\end_inset + + y +\begin_inset Formula $dN_{p}=-A_{p}\equiv0$ +\end_inset + +, luego +\begin_inset Formula $N$ +\end_inset + + es constante en algún +\begin_inset Formula $a\in\mathbb{R}^{3}$ +\end_inset + +. + Sean ahora +\begin_inset Formula $\phi(p):=\langle p,a\rangle$ +\end_inset + +, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + y +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +, entonces +\begin_inset Formula $d\phi_{p}(v)=\frac{d(\phi\circ\alpha)}{dt}(0)=\frac{d}{dt}(\langle\alpha(t),a\rangle)(0)=\langle\alpha'(0),a\rangle=\langle v,a\rangle\overset{a=N(p)}{=}0$ +\end_inset + +, luego +\begin_inset Formula $\phi$ +\end_inset + + es constante en algún +\begin_inset Formula $d\in\mathbb{R}$ +\end_inset + + y +\begin_inset Formula $S\subseteq\{\langle p,a\rangle=d\}=\{\langle p-p',a\rangle=0\}$ +\end_inset + + para algún +\begin_inset Formula $p'$ +\end_inset + + con +\begin_inset Formula $\langle p',a\rangle=d$ +\end_inset + +, pero +\begin_inset Formula $\{\langle p-p',a\rangle=0\}=p'+\langle a\rangle^{\bot}$ +\end_inset + +, luego +\begin_inset Formula $S$ +\end_inset + + esta contenido en un plano. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $c\neq0$ +\end_inset + +, sea +\begin_inset Formula $\phi:S\to\mathbb{R}^{3}$ +\end_inset + + la función diferenciable dada por +\begin_inset Formula $\phi(p):=p+\frac{1}{c}N(p)$ +\end_inset + +, para +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + y una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +, entonces +\begin_inset Formula +\begin{align*} +d\phi_{p}(v) & =\frac{d(\phi\circ\alpha)}{dt}(0)=\frac{d}{dt}\left(\alpha(t)+\frac{1}{c}N(\alpha(t))\right)(0)=\alpha'(0)+\frac{1}{c}(N\circ\alpha)'(0)\\ + & =v+\frac{1}{c}dN_{p}(v)=v-\frac{1}{c}A_{p}v=v-\frac{1}{c}cv=0, +\end{align*} + +\end_inset + +luego +\begin_inset Formula $\phi$ +\end_inset + + es constante en algún +\begin_inset Formula $a\in\mathbb{R}^{3}$ +\end_inset + +. + Pero para +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $p-a=-\frac{1}{c}N(p)$ +\end_inset + +, luego +\begin_inset Formula $\Vert p-a\Vert^{2}=\frac{1}{c^{2}}$ +\end_inset + + y todos los puntos de +\begin_inset Formula $S$ +\end_inset + + están en la esfera +\begin_inset Formula $a+\mathbb{S}^{2}(\frac{1}{c^{2}})$ +\end_inset + +. +\end_layout + +\begin_layout Section +Parámetros de la segunda forma fundamental +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada por +\begin_inset Formula $N$ +\end_inset + + y +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + +, los +\series bold +coeficientes de la segunda forma fundamental +\series default + son +\begin_inset Formula $e,f,g:U\to\mathbb{R}$ +\end_inset + + dados por +\begin_inset Formula +\begin{align*} +e & :=\langle N,X_{uu}\rangle=-\langle N_{u},X_{u}\rangle,\\ +f & :=\langle N,X_{uv}\rangle=-\langle N_{v},X_{u}\rangle=-\langle N_{u},X_{v}\rangle,\\ +g & :=\langle N,X_{vv}\rangle=-\langle N_{v},X_{v}\rangle, +\end{align*} + +\end_inset + +y para +\begin_inset Formula $p\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p}S$ +\end_inset + +, si +\begin_inset Formula $q:=X^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $v=v_{1}X_{u}(q)+v_{2}X_{v}(q)$ +\end_inset + +, entonces +\begin_inset Formula +\[ +{\cal II}_{p}(v):=v_{1}^{2}e+2v_{1}v_{2}f+v_{2}^{2}g. +\] + +\end_inset + + +\series bold +Demostración: +\series default + +\begin_inset Formula $\langle N,X_{u}\rangle=\langle N,X_{v}\rangle=0$ +\end_inset + +, y derivando se obtiene +\begin_inset Formula $\langle N_{u},X_{u}\rangle+\langle N,X_{uu}\rangle=0$ +\end_inset + +, +\begin_inset Formula $\langle N_{v},X_{u}\rangle+\langle N,X_{uv}\rangle=0$ +\end_inset + +, +\begin_inset Formula $\langle N_{u},X_{v}\rangle+\langle N,X_{vu}\rangle=0$ +\end_inset + + y +\begin_inset Formula $\langle N_{v},X_{v}\rangle+\langle N,X_{vv}\rangle=0$ +\end_inset + +, lo que nos da las igualdades en los coeficientes teniendo en cuenta que + +\begin_inset Formula $\langle N,X_{uv}\rangle=\langle N,X_{vu}\rangle$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $q:=X^{-1}(p)=(u(0),v(0))$ +\end_inset + +, por linealidad +\begin_inset Formula $dN_{p}(v)=v_{1}dN_{p}(X_{u}(q))+v_{2}dN_{p}(X_{v}(q))=v_{1}N_{u}(q)+v_{2}N_{v}(q)$ +\end_inset + +. + Entonces, evaluando las derivadas de +\begin_inset Formula $X$ +\end_inset + + y +\begin_inset Formula $N$ +\end_inset + + en +\begin_inset Formula $q$ +\end_inset + +, d +\begin_inset Formula +\begin{align*} +{\cal II}_{p}(v) & =\langle A_{p}v,v\rangle=-\langle dN_{p}(v),v\rangle=-\langle v_{1}N_{u}+v_{2}N_{v},v_{1}X_{u}+v_{2}X_{v}\rangle\\ + & =v_{1}^{2}\langle N_{u},X_{u}\rangle-v_{1}v_{2}\langle N_{u},X_{v}\rangle-v_{1}v_{2}\langle N_{v},X_{u}\rangle-v_{2}^{2}\langle N_{v},X_{v}\rangle\\ + & =v_{1}^{2}e+v_{1}v_{2}f+v_{2}^{2}g. +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula +\[ +dN_{p}\equiv\begin{pmatrix}a_{11} & a_{12}\\ +a_{21} & a_{22} +\end{pmatrix} +\] + +\end_inset + +respecto de la base +\begin_inset Formula $(X_{u},X_{v})$ +\end_inset + +, entonces +\begin_inset Formula +\begin{align*} +\begin{pmatrix}-e & -f\\ +-f & -g +\end{pmatrix} & =\begin{pmatrix}a_{11} & a_{21}\\ +a_{12} & a_{22} +\end{pmatrix}\begin{pmatrix}E & F\\ +F & G +\end{pmatrix} +\end{align*} + +\end_inset + + y tenemos las +\series bold +fórmulas de Weingarten: +\series default + +\begin_inset Formula +\begin{align*} +a_{11} & =\frac{fF-eG}{EG-F^{2}}, & a_{12} & =\frac{gF-fG}{EG-F^{2}}, & a_{21} & =\frac{eF-fE}{EG-F^{2}}, & a_{22} & =\frac{fF-gE}{EG-F^{2}}. +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + +\begin_inset Formula +\begin{align*} +-e & =\langle N_{u},X_{u}\rangle=\langle a_{11}X_{u}+a_{21}X_{v},X_{u}\rangle=a_{11}E+a_{21}F,\\ +-f & =\langle N_{v},X_{u}\rangle=\langle a_{12}X_{u}+a_{22}X_{v},X_{u}\rangle=a_{12}E+a_{22}F\\ + & =\langle N_{u},X_{v}\rangle=\langle a_{11}X_{u}+a_{21}X_{v},X_{v}\rangle=a_{11}F+a_{21}G,\\ +-g & =\langle N_{v},X_{v}\rangle=\langle a_{12}X_{u}+a_{22}X_{v},X_{v}\rangle=a_{12}F+a_{22}G. +\end{align*} + +\end_inset + +Despejando, +\begin_inset Formula +\[ +\begin{pmatrix}a_{11} & a_{12}\\ +a_{12} & a_{22} +\end{pmatrix}=-\begin{pmatrix}e & f\\ +f & g +\end{pmatrix}\begin{pmatrix}E & F\\ +F & G +\end{pmatrix}^{-1}=-\frac{1}{EG-F^{2}}\begin{pmatrix}e & f\\ +f & g +\end{pmatrix}\begin{pmatrix}G & -F\\ +-F & E +\end{pmatrix}, +\] + +\end_inset + +lo que nos da las fórmulas de Weingarten. +\end_layout + +\begin_layout Standard +De aquí, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align*} +K(p) & =\frac{eg-f^{2}}{EG-F^{2}}, & H(p) & =\frac{1}{2}\frac{eG+gE-2fF}{EG-F^{2}}, +\end{align*} + +\end_inset + +y las curvaturas principales son +\begin_inset Formula +\[ +\kappa_{i}(p)=H(p)\pm\sqrt{H(p)^{2}-K(p)}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + +\begin_inset Formula +\begin{align*} +K(p) & =\det A_{p}=\det(dN_{p})=\frac{1}{EG-F^{2}}((fF-eG)(fF-gE)-(gF-fG)(eF-fE))\\ + & =\frac{1}{(EG-F^{2})^{2}}(f^{2}F^{2}-fgEF-efFG+egEG-egF^{2}+fgEF+efFG-f^{2}EG)\\ + & =\frac{f^{2}F^{2}+egEG-egF^{2}-f^{2}EG}{(EG-F^{2})^{2}}=\frac{(EG-F^{2})(eg-f^{2})}{(EG-F^{2})^{2}},\\ +H(p) & =\frac{1}{2}\text{tr}A_{p}=-\frac{1}{2}\text{tr}(dN_{p})=-\frac{1}{2}\frac{2fF-eG-gE}{EG-F^{2}}. +\end{align*} + +\end_inset + +Un +\begin_inset Formula $\lambda\in\mathbb{R}$ +\end_inset + + es un valor propio de +\begin_inset Formula $A_{p}$ +\end_inset + + si y sólo si +\begin_inset Formula +\begin{align*} +0 & =\det(\lambda1_{T_{p}S}-A_{p})=\det(dN_{p}+\lambda1_{T_{p}S})=\begin{vmatrix}a_{11}+\lambda & a_{12}\\ +a_{21} & a_{22}+\lambda +\end{vmatrix}\\ + & =\lambda^{2}+(a_{11}+a_{22})\lambda+(a_{11}a_{22}-a_{12}a_{21})=\lambda^{2}-2H(p)+K(p), +\end{align*} + +\end_inset + +si y sólo si +\begin_inset Formula $\lambda=H(p)\pm\sqrt{H(p)^{2}-K(p)}$ +\end_inset + +. +\end_layout + +\begin_layout Section +Isometrías locales +\end_layout + +\begin_layout Standard +Una +\series bold +isometría local +\series default + entre dos superficies regulares +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + es una función diferenciable +\begin_inset Formula $\phi:S_{1}\to S_{2}$ +\end_inset + + tal que para +\begin_inset Formula $p\in S_{1}$ +\end_inset + + y +\begin_inset Formula $v,w\in T_{p}S_{1}$ +\end_inset + + es +\begin_inset Formula $\langle d\phi_{p}(v),d\phi_{p}(w)\rangle=\langle v,w\rangle$ +\end_inset + +, es decir, tal que +\begin_inset Formula $d\phi_{p}:T_{p}S_{1}\to T_{\phi(p)}S_{2}$ +\end_inset + + es una isometría lineal. + Entonces +\begin_inset Formula $\phi$ +\end_inset + + conserva ángulos, longitudes y áreas de +\begin_inset Formula $S_{1}$ +\end_inset + + a +\begin_inset Formula $S_{2}$ +\end_inset + +, pero su existencia no implica que exista una isometría lineal +\begin_inset Formula $\psi:S_{2}\to S_{1}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Una +\series bold +isometría +\series default + ( +\series bold +global +\series default +) entre +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + es una isometría local que es un difeomorfismo. + +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + son ( +\series bold +globalmente +\series default +) +\series bold +isométricas +\series default + si existe una isometría global entre ellas, y son +\series bold +localmente isométricas +\series default + si para cada +\begin_inset Formula $p\in S_{1}$ +\end_inset + + hay un entorno +\begin_inset Formula $V\subseteq S_{1}$ +\end_inset + + de +\begin_inset Formula $p$ +\end_inset + + y una isometría global +\begin_inset Formula $\phi:V\to\phi(V)\subseteq S_{2}$ +\end_inset + + y para cada +\begin_inset Formula $q\in S_{2}$ +\end_inset + + hay un entorno +\begin_inset Formula $W\subseteq S_{2}$ +\end_inset + + de +\begin_inset Formula $p$ +\end_inset + + y una isometría global +\begin_inset Formula $\psi:W\to\phi(W)\subseteq S_{1}$ +\end_inset + +. + Si existe una isometría local entre +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + +, +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + son localmente isométricos. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{TS} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $(\pi_{1}(X,x),*)$ +\end_inset + + es un grupo, llamado +\series bold +grupo fundamental +\series default + [...] de +\begin_inset Formula $X$ +\end_inset + + relativo al +\series bold +punto base +\series default + +\begin_inset Formula $x$ +\end_inset + + [...] +\begin_inset Formula $X$ +\end_inset + + es +\series bold +simplemente conexo +\series default + si es conexo por caminos y +\begin_inset Formula $\pi_{1}(X,x)$ +\end_inset + + es el grupo trivial [...] para todo +\begin_inset Formula $x\in X$ +\end_inset + +. + [...] Todo subespacio estrellado de +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + + es simplemente conexo. + [...] El grupo fundamental de +\begin_inset Formula $\mathbb{S}^{1}$ +\end_inset + + es isomorfo a +\begin_inset Formula $(\mathbb{Z},+)$ +\end_inset + +. + [...] +\begin_inset Formula $\pi_{1}(X\times Y,(x,y))\cong\pi_{1}(X,x)\times\pi_{1}(Y,y)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Existe una isometría local entre el plano +\begin_inset Formula $\Pi:=\{z=0\}$ +\end_inset + + y el cilindro +\begin_inset Formula $C:=\mathbb{S}^{1}\times\mathbb{R}$ +\end_inset + +, pero las superficies no son globalmente isométricas. + +\series bold +Demostración: +\series default + Como el plano es estrellado, su grupo fundamental es el grupo trivial, + y como el cilindro es +\begin_inset Formula $\mathbb{S}^{1}\times\mathbb{R}$ +\end_inset + +, su grupo fundamental es +\begin_inset Formula $\pi_{1}(\mathbb{S}^{1}\times\mathbb{R},e_{1})\cong\pi_{1}(\mathbb{S}_{1},e_{1})\times\pi_{1}(\mathbb{R},0)\cong(\mathbb{Z},+)\times1\cong(\mathbb{Z},+)$ +\end_inset + +. + Como los grupos fundamentales no son isomorfos, +\begin_inset Formula $\Pi$ +\end_inset + + y +\begin_inset Formula $C$ +\end_inset + + no son homeomorfos y por tanto tampoco isométricos. + Sea ahora +\begin_inset Formula $\phi:\Pi\to C$ +\end_inset + + dada por +\begin_inset Formula $\phi(x,y,0):=(\cos x,\sin x,y)$ +\end_inset + +, que es diferenciable. + Para +\begin_inset Formula $p=(x,y,0)\in\Pi$ +\end_inset + +, +\begin_inset Formula $T_{p}S=\Pi$ +\end_inset + +, y si +\begin_inset Formula $v=(v_{1},v_{2},0)\in T_{p}S$ +\end_inset + +, sea +\begin_inset Formula $\alpha:I\to\Pi$ +\end_inset + + dada por +\begin_inset Formula $\alpha(t):=p+tv$ +\end_inset + +, +\begin_inset Formula +\[ +d\phi_{p}(v)=\frac{d(\phi\circ\alpha)}{dt}(0)=\frac{d}{dt}(\cos(x+tv_{1}),\sin(x+tv_{1}),y+tv_{2})(0)=(-v_{1}\sin x,v_{1}\cos x,v_{2}). +\] + +\end_inset + +Para ver que +\begin_inset Formula $\phi$ +\end_inset + + conserva el producto escalar, basta ver que conserva módulos, pero +\begin_inset Formula $|d\phi_{p}(v)|^{2}=v_{1}^{2}+v_{2}^{2}=|v|^{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, sea +\begin_inset Formula $\phi:S_{1}\to S_{2}$ +\end_inset + + una isometría local entre superficies regulares, para todo +\begin_inset Formula $p\in S_{1}$ +\end_inset + + existen parametrizaciones +\begin_inset Formula $(U,X)$ +\end_inset + + de +\begin_inset Formula $S_{1}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $(U,\overline{X})$ +\end_inset + + de +\begin_inset Formula $S_{2}$ +\end_inset + + en +\begin_inset Formula $\phi(p)$ +\end_inset + + con los mismos parámetros de la primera forma fundamental. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $(\tilde{U},X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S_{1}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $\overline{X}:\phi\circ X:\tilde{U}\to S_{2}$ +\end_inset + +, como +\begin_inset Formula $\phi$ +\end_inset + + es un difeomorfismo local, existe un entorno +\begin_inset Formula $V\subseteq S_{1}$ +\end_inset + + de +\begin_inset Formula $p$ +\end_inset + + en el que +\begin_inset Formula $\phi:V\to\phi(V)$ +\end_inset + + es un difeomorfismo, por lo que si +\begin_inset Formula $U:=X^{-1}(V)\subseteq\tilde{U}$ +\end_inset + +, restringiendo +\begin_inset Formula $\overline{X}$ +\end_inset + + a +\begin_inset Formula $U$ +\end_inset + +, +\begin_inset Formula $(U,\overline{X})$ +\end_inset + + es una parametrización de +\begin_inset Formula $S_{2}$ +\end_inset + + en +\begin_inset Formula $\phi(p)$ +\end_inset + +. + Entonces, si +\begin_inset Formula $q:=X^{-1}(p)$ +\end_inset + +, +\begin_inset Formula $d\overline{X}_{q}=d(\phi\circ X)_{q}=d\phi_{p}\circ dX_{q}$ +\end_inset + +, luego +\begin_inset Formula $\overline{X}_{u}(q)=d\phi_{p}(X_{u}(q))$ +\end_inset + + y +\begin_inset Formula $\overline{X}_{v}(q)=d\phi_{p}(X_{v}(q))$ +\end_inset + +. + Con esto, como +\begin_inset Formula $\phi$ +\end_inset + + es una isometría local, +\begin_inset Formula $\overline{E}=\langle\overline{X}_{u}(q),\overline{X}_{u}(q)\rangle=\langle d\phi_{p}(X_{u}(q)),d\phi(X_{u}(q))\rangle=\langle X_{u}(q),X_{u}(q)\rangle=E$ +\end_inset + +, y análogamente +\begin_inset Formula $\overline{F}=F$ +\end_inset + + y +\begin_inset Formula $\overline{G}=G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, dadas dos superficies regulares +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + y dos parametrizaciones +\begin_inset Formula $(U,X)$ +\end_inset + + de +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $(U,\overline{X})$ +\end_inset + + de +\begin_inset Formula $S_{2}$ +\end_inset + + con los mismos parámetros de la primera forma fundamental, entonces +\begin_inset Formula $\phi:=\overline{X}\circ X^{-1}:X(U)\to\overline{X}(U)$ +\end_inset + + es una isometría. + +\series bold +Demostración: +\series default + Es un difeomorfismo por ser composición de difeomorfismos, y queda ver + que conserva productos escalares. + Sean +\begin_inset Formula $q\in U$ +\end_inset + + y +\begin_inset Formula $p:=X(q)$ +\end_inset + +, +\begin_inset Formula $d\phi_{p}\circ dX_{q}=d(\phi\circ X)_{q}=d\overline{X}_{q}$ +\end_inset + + por la regla de la cadena, por lo que +\begin_inset Formula $d\phi_{p}(X_{u}(q))=\overline{X}_{u}(q)$ +\end_inset + + y +\begin_inset Formula $d\phi_{p}(X_{v}(q))=\overline{X}_{v}(q)$ +\end_inset + +. + Por tanto, en +\begin_inset Formula $q$ +\end_inset + +, +\begin_inset Formula $\langle d\phi_{p}(X_{u}),d\phi_{p}(X_{u})\rangle=\langle\overline{X}_{u},\overline{X}_{u}\rangle=\overline{E}=E=\langle X_{u},X_{u}\rangle$ +\end_inset + +, y de forma análoga +\begin_inset Formula $\langle d\phi_{p}(X_{u}),d\phi_{p}(X_{v})\rangle=\langle X_{u},X_{v}\rangle$ +\end_inset + + y +\begin_inset Formula $\langle d\phi_{p}(X_{v}),d\phi_{p}(X_{v})\rangle=\langle X_{v},X_{v}\rangle$ +\end_inset + +, pero +\begin_inset Formula $(X_{u},X_{v})$ +\end_inset + + es una base de +\begin_inset Formula $T_{p}S$ +\end_inset + +, luego +\begin_inset Formula $d\phi_{p}$ +\end_inset + + conserva productos escalares. +\end_layout + +\begin_layout Section + +\lang latin +Theorema Egregium +\lang spanish + de Gauss +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada por +\begin_inset Formula $N$ +\end_inset + + y +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + con la base +\begin_inset Formula $(X_{u},X_{v},N)$ +\end_inset + + de +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + + positivamente orientada. + Las +\series bold +fórmulas de Gauss +\series default + son +\begin_inset Formula +\[ +\left\{ \begin{aligned}X_{uu} & =\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN,\\ +X_{uv} & =\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN,\\ +X_{vu} & =\Gamma_{21}^{1}X_{u}+\Gamma_{21}^{2}X_{v}+fN,\\ +X_{vv} & =\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN, +\end{aligned} +\right. +\] + +\end_inset + +donde los +\begin_inset Formula $\Gamma_{ij}^{k}$ +\end_inset + + son los +\series bold +símbolos de Christoffel +\series default +, y se basan en que +\begin_inset Formula $\langle X_{uu},N\rangle=e$ +\end_inset + +, +\begin_inset Formula $\langle X_{uv},N\rangle=\langle X_{vu},N\rangle=f$ +\end_inset + + y +\begin_inset Formula $\langle X_{vv},N\rangle=g$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Gamma_{12}^{1}=\Gamma_{21}^{1}$ +\end_inset + + y +\begin_inset Formula $\Gamma_{12}^{2}=\Gamma_{22}^{2}$ +\end_inset + +, pues +\begin_inset Formula $X_{uv}=X_{vu}$ +\end_inset + +. + Además, +\begin_inset Formula +\[ +\begin{pmatrix}\Gamma_{11}^{1} & \Gamma_{12}^{1} & \Gamma_{22}^{1}\\ +\Gamma_{11}^{2} & \Gamma_{12}^{2} & \Gamma_{22}^{2} +\end{pmatrix}=\frac{1}{EG-F^{2}}\begin{pmatrix}G & -F\\ +-F & E +\end{pmatrix}\begin{pmatrix}\frac{E_{u}}{2} & \frac{E_{v}}{2} & F_{v}-\frac{G_{u}}{2}\\ +F_{u}-\frac{E_{v}}{2} & \frac{G_{u}}{2} & \frac{G_{v}}{2} +\end{pmatrix}. +\] + +\end_inset + + +\series bold +Demostración: +\series default + Multiplicando escalarmente las ecuaciones de Gauss por +\begin_inset Formula $X_{u}$ +\end_inset + + y +\begin_inset Formula $X_{v}$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +\langle X_{uu},X_{u}\rangle & =\Gamma_{11}^{1}E+\Gamma_{11}^{2}F, & \langle X_{uu},X_{v}\rangle & =\Gamma_{11}^{1}F+\Gamma_{11}^{2}G,\\ +\langle X_{uv},X_{u}\rangle & =\Gamma_{12}^{1}E+\Gamma_{12}^{2}F, & \langle X_{uv},X_{v}\rangle & =\Gamma_{12}^{1}F+\Gamma_{12}^{2}G,\\ +\langle X_{vv},X_{u}\rangle & =\Gamma_{22}^{1}E+\Gamma_{22}^{2}F, & \langle X_{vv},X_{v}\rangle & =\Gamma_{22}^{1}F+\Gamma_{22}^{2}G. +\end{align*} + +\end_inset + +Derivando +\begin_inset Formula $E$ +\end_inset + +, +\begin_inset Formula $F$ +\end_inset + + y +\begin_inset Formula $G$ +\end_inset + + respecto a +\begin_inset Formula $u$ +\end_inset + + y +\begin_inset Formula $v$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +E_{u} & =2\langle X_{uu},X_{u}\rangle, & F_{u} & =\langle X_{uu},X_{v}\rangle+\langle X_{u},X_{vu}\rangle, & G_{u} & =2\langle X_{vu},X_{v}\rangle,\\ +E_{v} & =2\langle X_{uv},X_{u}\rangle, & F_{v} & =\langle X_{uv},X_{v}\rangle+\langle X_{u},X_{vv}\rangle, & G_{v} & =2\langle X_{vv},X_{v}\rangle, +\end{align*} + +\end_inset + +por lo que +\begin_inset Formula +\begin{align*} +\langle X_{uu},X_{u}\rangle & =\frac{E_{u}}{2}, & \langle X_{uv},X_{u}\rangle & =\frac{E_{v}}{2}, & \langle X_{vv},X_{u}\rangle & =F_{v}-\langle X_{uv},X_{v}\rangle=F_{v}-\frac{G_{u}}{2},\\ +\langle X_{uv},X_{v}\rangle & =\frac{G_{u}}{2}, & \langle X_{vv},X_{v}\rangle & =\frac{G_{v}}{2}, & \langle X_{uu},X_{v}\rangle & =F_{u}-\langle X_{u},X_{vu}\rangle=F_{u}-\frac{E_{v}}{2}. +\end{align*} + +\end_inset + +Igualando queda el sistema +\begin_inset Formula +\[ +\left\{ \begin{aligned}E\Gamma_{11}^{1}+F\Gamma_{11}^{2} & =\frac{1}{2}E_{u}, & E\Gamma_{12}^{1}+F\Gamma_{12}^{2} & =\frac{1}{2}E_{v}, & E\Gamma_{22}^{1}+F\Gamma_{22}^{2} & =F_{v}-\frac{1}{2}G_{u},\\ +F\Gamma_{11}^{1}+G\Gamma_{11}^{2} & =F_{u}-\frac{1}{2}E_{v}, & F\Gamma_{12}^{1}+G\Gamma_{12}^{2} & =\frac{1}{2}G_{u}, & F\Gamma_{22}^{1}+G\Gamma_{22}^{2} & =\frac{1}{2}G_{v}, +\end{aligned} +\right. +\] + +\end_inset + +que se divide en tres sistemas disjuntos de izquierda a derecha. + Para el primero, +\begin_inset Formula +\[ +\begin{pmatrix}\Gamma_{11}^{1}\\ +\Gamma_{12}^{2} +\end{pmatrix}=\begin{pmatrix}E & F\\ +F & G +\end{pmatrix}^{-1}\begin{pmatrix}\frac{1}{2}E_{u}\\ +F_{u}-\frac{E_{v}}{2} +\end{pmatrix}=\frac{1}{EG-F^{2}}\begin{pmatrix}G & -F\\ +-F & E +\end{pmatrix}\begin{pmatrix}\frac{1}{2}E_{u}\\ +F_{u}-\frac{E_{v}}{2} +\end{pmatrix}, +\] + +\end_inset + +y para los otros dos es análogo. +\end_layout + +\begin_layout Standard +La +\series bold +ecuación de Gauss +\series default + es +\begin_inset Formula +\[ +\Gamma_{11}^{1}\Gamma_{12}^{2}+(\Gamma_{11}^{2})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{2}-\Gamma_{12}^{1}\Gamma_{11}^{2}-(\Gamma_{12}^{2})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{2}=EK, +\] + +\end_inset + +la primera +\series bold +ecuación de Mainardi-Codazzi +\series default + es +\begin_inset Formula +\[ +e\Gamma_{12}^{1}+f(\Gamma_{12}^{2}-\Gamma_{11}^{1})-g\Gamma_{11}^{2}=e_{v}-f_{u} +\] + +\end_inset + +y, además, +\begin_inset Formula +\begin{align*} +(\Gamma_{11}^{1})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{1}-(\Gamma_{12}^{1})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{1} & =-FK. +\end{align*} + +\end_inset + + +\series bold +Demostración: +\series default + +\begin_inset Formula $X_{uuv}=X_{uvu}$ +\end_inset + +, y sustituyendo +\begin_inset Formula $X_{uu}$ +\end_inset + + y +\begin_inset Formula $X_{vv}$ +\end_inset + + según las fórmulas de Gauss, +\begin_inset Formula +\begin{multline*} +0=X_{uuv}-X_{uvu}=(\Gamma_{11}^{1})_{v}X_{u}+\Gamma_{11}^{1}X_{uv}+(\Gamma_{11}^{2})_{v}X_{v}+\Gamma_{11}^{2}X_{vv}+e_{v}N+eN_{v}-\\ +-(\Gamma_{12}^{1})_{u}X_{u}-\Gamma_{12}^{1}X_{uu}-(\Gamma_{12}^{2})_{u}X_{v}-\Gamma_{12}^{2}X_{vu}-f_{u}N-fN_{u}. +\end{multline*} + +\end_inset + +Sustituyendo con las fórmulas de Gauss, +\begin_inset Formula +\begin{multline*} +0=(\Gamma_{11}^{1})_{v}X_{u}+\Gamma_{11}^{1}(\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN)+(\Gamma_{11}^{2})_{v}X_{v}+\Gamma_{11}^{2}(\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN)-\\ +-(\Gamma_{12}^{1})_{u}X_{u}-\Gamma_{12}^{1}(\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN)-(\Gamma_{12}^{2})_{u}X_{v}-\Gamma_{12}^{2}(\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN)+\\ ++e_{v}N+e(a_{12}X_{u}+a_{22}X_{v})-f_{u}N-f(a_{11}X_{u}+a_{21}X_{v})=:A_{1}X_{u}+B_{1}X_{v}+C_{1}N. +\end{multline*} + +\end_inset + +Como +\begin_inset Formula $(X_{u},X_{v},N)$ +\end_inset + + es base de +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + +, +\begin_inset Formula $A_{1},B_{1},C_{1}=0$ +\end_inset + +. + Como +\begin_inset Formula $B_{1}=0$ +\end_inset + +, usando las fórmulas de Weingarten, +\begin_inset Formula +\begin{multline*} +\Gamma_{11}^{1}\Gamma_{12}^{2}+(\Gamma_{11}^{2})_{v}X_{v}+\Gamma_{11}^{2}\Gamma_{22}^{2}-\Gamma_{12}^{1}\Gamma_{11}^{2}-(\Gamma_{12}^{2})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{2}=fa_{21}-ea_{22}=\\ +=f\frac{eF-fE}{EG-F^{2}}-e\frac{fF-gE}{EG-F^{2}}=\frac{efF-f^{2}E-efF+egE}{EG-F^{2}}=E\frac{eg-f^{2}}{EG-F^{2}}=EK. +\end{multline*} + +\end_inset + + +\begin_inset Formula $C_{1}=0$ +\end_inset + + nos da +\begin_inset Formula +\begin{multline*} +\Gamma_{11}^{1}f+\Gamma_{11}^{2}g-\Gamma_{12}^{1}e-\Gamma_{12}^{1}f+e_{v}-f_{u}=0, +\end{multline*} + +\end_inset + +de donde se obtiene directamente la primera ecuación de Mainardi-Codazzi, + y +\begin_inset Formula $A_{1}=0$ +\end_inset + + nos da +\begin_inset Formula +\begin{multline*} +(\Gamma_{11}^{1})_{v}+\Gamma_{11}^{1}\Gamma_{12}^{1}+\Gamma_{11}^{2}\Gamma_{22}^{1}-(\Gamma_{12}^{1})_{u}-\Gamma_{12}^{1}\Gamma_{11}^{1}-\Gamma_{12}^{2}\Gamma_{12}^{1}=(\Gamma_{11}^{1})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{1}-(\Gamma_{12}^{1})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{1}=\\ +=fa_{11}-ea_{12}=f\frac{fF-eG}{EG-F^{2}}-e\frac{gF-fG}{EG-F^{2}}=\frac{f^{2}F-egF}{EG-F^{2}}=-FK. +\end{multline*} + +\end_inset + + +\end_layout + +\begin_layout Standard +La curvatura de Gauss depende solo de la primera forma fundamental, pues + como +\begin_inset Formula $EG-F^{2}>0$ +\end_inset + +, +\begin_inset Formula $E\neq0$ +\end_inset + + y por la ecuación de Gauss +\begin_inset Formula $K$ +\end_inset + + se puede obtener de +\begin_inset Formula $E$ +\end_inset + + y los símbolos de Christoffel, que dependen solo de la primera forma fundamenta +l. +\end_layout + +\begin_layout Standard + +\series bold +\lang latin +Theorema Egregium +\lang spanish + de Gauss: +\series default + La curvatura de Gauss de una superficie regular es invariante por isometrías + locales. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $\phi:S_{1}\to S_{2}$ +\end_inset + + una isometría local entre superficies regulares, +\begin_inset Formula $p\in S_{1}$ +\end_inset + + y +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S_{1}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + con +\begin_inset Formula $U$ +\end_inset + + lo suficientemente pequeña para que +\begin_inset Formula $\phi|_{V:=X(U)}:V\to\phi(V)$ +\end_inset + + sea un difeomorfismo, entonces +\begin_inset Formula $(U,\overline{X}:=\phi\circ X)$ +\end_inset + + es una parametrización de +\begin_inset Formula $S_{2}$ +\end_inset + + en +\begin_inset Formula $\phi(p)$ +\end_inset + +. + Entonces, como los coeficientes de la primera forma fundamental son los + mismos para ambas parametrizaciones y la curvatura de Gauss solo depende + de estos, las curvaturas de Gauss coinciden para el mismo punto de +\begin_inset Formula $U$ +\end_inset + + y en particular +\begin_inset Formula $K_{1}(p)=K_{2}(\phi(p))$ +\end_inset + +, donde +\begin_inset Formula $K_{1}$ +\end_inset + + y +\begin_inset Formula $K_{2}$ +\end_inset + + son las curvaturas de Gauss respectivas de +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +En general un difeomorfismo local que conserva la curvatura no es una isometría + local. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + parametrizadas por +\begin_inset Formula $X(u,v):=(u\cos v,u\sin v,\log u)$ +\end_inset + + y +\begin_inset Formula $\overline{X}(u,v):=(u\cos v,u\sin v,v)$ +\end_inset + +, entonces +\begin_inset Formula +\begin{align*} +X_{u} & =(\cos v,\sin v,\tfrac{1}{u}), & \overline{X}_{u} & =(\cos v,\sin v,0),\\ +X_{v} & =(-u\sin v,u\cos v,0), & \overline{X}_{v} & =(-u\sin v,u\cos v,1),\\ +N & =\frac{(-\cos v,-\sin v,u)}{\sqrt{1+u^{2}}}, & \overline{N} & =\frac{(\sin v,-\cos v,u)}{\sqrt{1+u^{2}}}, +\end{align*} + +\end_inset + +luego +\begin_inset Formula $N$ +\end_inset + + y +\begin_inset Formula $\overline{N}$ +\end_inset + + se diferencian en alguna transformación ortogonal. + Si +\begin_inset Formula $\overline{N}=O\circ N$ +\end_inset + + para una transformación ortogonal +\begin_inset Formula $O$ +\end_inset + +, entonces +\begin_inset Formula $d\overline{N}_{q}=dO_{N(q)}\circ dN_{q}=O\circ dN_{q}$ +\end_inset + +, luego +\begin_inset Formula $d\overline{N}_{q}$ +\end_inset + + y +\begin_inset Formula $dN_{q}$ +\end_inset + + se diferencian por +\begin_inset Formula $O$ +\end_inset + + y por tanto tienen igual determinante, que será la curvatura de Gauss. + Sin embargo, +\begin_inset Formula $\phi:=\overline{X}\circ X^{-1}=((x,y,z)\mapsto(x,y,e^{z}))$ +\end_inset + + no es una isometría. +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align*} +a_{11} & =\frac{fF-eG}{EG-F^{2}}, & a_{12} & =\frac{gF-fG}{EG-F^{2}}, & a_{21} & =\frac{eF-fE}{EG-F^{2}}, & a_{22} & =\frac{fF-gE}{EG-F^{2}}. +\end{align*} + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +La segunda +\series bold +ecuación de Mainardi-Codazzi +\series default + es +\begin_inset Formula +\[ +f_{v}-g_{u}=e\Gamma_{22}^{1}+f(\Gamma_{22}^{2}-\Gamma_{12}^{1})-g\Gamma_{12}^{2}. +\] + +\end_inset + + +\series bold +Demostración: +\series default + Como +\begin_inset Formula $X_{vvu}=X_{vuv}$ +\end_inset + +, aplicando las fórmulas de Gauss, +\begin_inset Note Comment +status open + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $gN_{u}$ +\end_inset + + is not Unix. +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{multline*} +0=X_{vvu}-X_{vuv}=(\Gamma_{22}^{1})_{u}X_{u}+\Gamma_{22}^{1}X_{uu}+(\Gamma_{22}^{2})_{u}X_{v}+\Gamma_{22}^{2}X_{vu}+g_{u}N+gN_{u}-\\ +-(\Gamma_{21}^{1})_{v}X_{u}-\Gamma_{21}^{1}X_{uv}-(\Gamma_{21}^{2})_{v}X_{v}-\Gamma_{21}^{2}X_{vv}-f_{v}N-fN_{v}, +\end{multline*} + +\end_inset + +y sustituyendo de nuevo, +\begin_inset Formula +\begin{multline*} +0=(\Gamma_{22}^{1})_{u}X_{u}+\Gamma_{22}^{1}(\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN)+(\Gamma_{22}^{2})_{u}X_{v}+\Gamma_{22}^{2}(\Gamma_{12}^{2}X_{u}+\Gamma_{12}^{2}X_{v}+fN)-\\ +-(\Gamma_{12}^{1})_{v}X_{u}-\Gamma_{12}^{1}(\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN)-(\Gamma_{12}^{2})_{v}X_{v}-\Gamma_{12}^{2}(\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN)+\\ ++g_{u}N+g(a_{11}X_{u}+a_{21}X_{v})-f_{v}N-f(a_{12}X_{u}+a_{22}X_{v})=:A_{2}X_{u}+B_{2}X_{v}+C_{2}N. +\end{multline*} + +\end_inset + +Como antes, +\begin_inset Formula $A_{2},B_{2},C_{2}=0$ +\end_inset + +, luego como +\begin_inset Formula $C_{2}=0$ +\end_inset + +, +\begin_inset Formula $e\Gamma_{22}^{1}+f\Gamma_{22}^{2}-f\Gamma_{12}^{1}-g\Gamma_{12}^{2}=f_{v}-g_{u}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Las +\series bold +ecuaciones de compatibilidad +\series default + son la ecuación de Gauss y las dos ecuaciones de Mainardi-Codazzi. + +\series bold +Teorema de Bonnet: +\series default + Sean +\begin_inset Formula $E,F,G,e,f,g:V\to\mathbb{R}$ +\end_inset + + funciones diferenciables en un abierto +\begin_inset Formula $V\subseteq\mathbb{R}^{2}$ +\end_inset + + con +\begin_inset Formula $E>0$ +\end_inset + +, +\begin_inset Formula $G>0$ +\end_inset + +, +\begin_inset Formula $EG-F^{2}>0$ +\end_inset + + y que verifican las ecuaciones de compatibilidad, entonces existen un abierto + +\begin_inset Formula $U\subseteq V$ +\end_inset + + y un difeomorfismo +\begin_inset Formula $X:U\to X(U)\subseteq\mathbb{R}^{3}$ +\end_inset + + tales que +\begin_inset Formula $(U,X)$ +\end_inset + + es una parametrización de la superficie regular +\begin_inset Formula $X(U)$ +\end_inset + + en la que los coeficientes de la primera y segunda formas fundamentales + son +\begin_inset Formula $E,F,G$ +\end_inset + + y +\begin_inset Formula $e,f,g$ +\end_inset + +, respectivamente, y si +\begin_inset Formula $U$ +\end_inset + + es conexo y +\begin_inset Formula $\overline{X}:U\to\overline{X}(U)$ +\end_inset + + es otro difeomorfismo con los mismos coeficientes de las formas fundamentales + primera y segunda, entonces existe un movimiento rígido +\begin_inset Formula $M$ +\end_inset + + en +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + + tal que +\begin_inset Formula $\overline{X}=M\circ X$ +\end_inset + +. +\end_layout + +\end_body +\end_document |